Représentations galoisiennes diédrales et formes à multiplication complexe

Abstract

Pour une représentation galoisienne diédrale en caractéristique ℓ on établit (sous certaines hypothèses) l'existence d'une newform à multiplication complexe, dont on contrôle le poids, le niveau et le caractère, telle que la représentation ℓ-adique associée est congrue modulo ℓ à celle de départ.Given a dihedral Galois representation in characteristic ℓ, we establish (under some assumption) the existence of a CM newform, whose weight, level and Nebentypus we pin down, such that its ℓ-adic representation is congruent modulo ℓ to the one we started with.

Examples of abelian surfaces failing the local-global principle for isogenies

“…corresponds to the imaginary quadratic field Q( √ −ℓ). We may now apply Théorème 1.1 of [BNMMdC18] to obtain the existence of a CM newform g such that ρ is isomorphic to the mod-λ ′ reduction of the λ ′ -adic G Qrepresentation ρ g,λ ′ , for some prime ideal λ ′ lying over ℓ in the Fourier coefficient field of g (which need not be the same as that of f ). Moreover, from the proof of Corollaire 1.3 in loc.…”